Maksimov and Kolovsky, Equation (3)

Average Error: 18.4 → 8.2

Time: 15.3s

Precision: binary64

Cost: 20352

Math

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)\right) \cdot \left(-2 \cdot J\right) \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (*
  (* (* -2.0 J) (cos (/ K 2.0)))
  (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))

↓

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* t_0 (hypot 1.0 (/ U (* J (* 2.0 t_0))))) (* -2.0 J))))

C

double code(double J, double K, double U) {
	return ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + pow((U / ((2.0 * J) * cos((K / 2.0)))), 2.0)));
}

↓

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return (t_0 * hypot(1.0, (U / (J * (2.0 * t_0))))) * (-2.0 * J);
}

Java

public static double code(double J, double K, double U) {
	return ((-2.0 * J) * Math.cos((K / 2.0))) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * Math.cos((K / 2.0)))), 2.0)));
}

↓

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return (t_0 * Math.hypot(1.0, (U / (J * (2.0 * t_0))))) * (-2.0 * J);
}

Python

def code(J, K, U):
	return ((-2.0 * J) * math.cos((K / 2.0))) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * math.cos((K / 2.0)))), 2.0)))

↓

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return (t_0 * math.hypot(1.0, (U / (J * (2.0 * t_0))))) * (-2.0 * J)

Julia

function code(J, K, U)
	return Float64(Float64(Float64(-2.0 * J) * cos(Float64(K / 2.0))) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * cos(Float64(K / 2.0)))) ^ 2.0))))
end

↓

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(t_0 * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0))))) * Float64(-2.0 * J))
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + ((U / ((2.0 * J) * cos((K / 2.0)))) ^ 2.0)));
end

↓

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = (t_0 * hypot(1.0, (U / (J * (2.0 * t_0))))) * (-2.0 * J);
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 18.4

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

2. Simplified 8.2

   \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)} \]
   Proof
   (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (hypot.f64 1 (/.f64 U (*.f64 J (*.f64 2 (cos.f64 (/.f64 K 2)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (hypot.f64 1 (/.f64 U (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 J 2) (cos.f64 (/.f64 K 2)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (hypot.f64 1 (/.f64 U (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 J)) (cos.f64 (/.f64 K 2))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))))))))): 36 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 -2 J) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))): 3 points increase in error, 10 points decrease in error

3. Final simplification 8.2

   \[\leadsto \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right) \cdot \left(-2 \cdot J\right) \]

Alternatives

Alternative 1
Error	17.3
Cost	14224

\[\begin{array}{l} t_0 := \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{if}\;J \leq -1.55 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 3.6 \cdot 10^{-199}:\\ \;\;\;\;-2 \cdot \left(J \cdot \frac{J}{U}\right) - U\\ \mathbf{elif}\;J \leq 8.8 \cdot 10^{-187}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 6.8 \cdot 10^{-161}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	26.0
Cost	7640

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{if}\;J \leq -1 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -1 \cdot 10^{-190}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.35 \cdot 10^{-198}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 4.6 \cdot 10^{-187}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.6 \cdot 10^{-154}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	21.0
Cost	7568

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{if}\;K \leq -1.1 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq -7.4 \cdot 10^{+97}:\\ \;\;\;\;U\\ \mathbf{elif}\;K \leq -0.06:\\ \;\;\;\;t_0\\ \mathbf{elif}\;K \leq 9.8 \cdot 10^{+29}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	38.8
Cost	1356

\[\begin{array}{l} \mathbf{if}\;J \leq -2.1 \cdot 10^{+182}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -2.5 \cdot 10^{+157}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.2 \cdot 10^{-10}:\\ \;\;\;\;J \cdot \left(-2 \cdot \left(1 + 0.125 \cdot \left(\frac{U}{J} \cdot \frac{U}{J}\right)\right)\right)\\ \mathbf{elif}\;J \leq -1.45 \cdot 10^{-192}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.02 \cdot 10^{-197}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.75 \cdot 10^{-186}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.9 \cdot 10^{-152}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 5 \cdot 10^{-16}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 5
Error	38.7
Cost	1248

\[\begin{array}{l} \mathbf{if}\;J \leq -8.5 \cdot 10^{+184}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1.15 \cdot 10^{+162}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -7 \cdot 10^{-15}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -7.8 \cdot 10^{-191}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.4 \cdot 10^{-198}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 5.5 \cdot 10^{-187}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3 \cdot 10^{-152}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 6
Error	38.8
Cost	1248

\[\begin{array}{l} \mathbf{if}\;J \leq -2.1 \cdot 10^{+182}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1.15 \cdot 10^{+162}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -1.56 \cdot 10^{-12}:\\ \;\;\;\;-2 \cdot J + \frac{U}{J} \cdot \left(U \cdot -0.25\right)\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-186}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9 \cdot 10^{-199}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 7.8 \cdot 10^{-187}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.2 \cdot 10^{-152}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3.3 \cdot 10^{-18}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 7
Error	46.8
Cost	920

\[\begin{array}{l} \mathbf{if}\;J \leq -3.3 \cdot 10^{-188}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.8 \cdot 10^{-198}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.2 \cdot 10^{-186}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.6 \cdot 10^{-153}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 4.8 \cdot 10^{+96}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.4 \cdot 10^{+248}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 8
Error	46.8
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022331 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))